[Python-Dev] Comparing heterogeneous types
Chermside, Michael
mchermside at ingdirect.com
Fri Jun 18 11:08:33 EDT 2004
Armin Rigo writes:
> Is it really what we want? It seems to me that the precision
> of a float gives
> roughly the magnitude of "uncertainty" of the value that it
> represents. If a
> float is large enough, then the same float can represent
> several long integer
> values. It should then probably be considered equal to any
> of these integers,
> instead of just equal to the one arbitrarily returned by
> casting it to long.
I think that what you describe is NOT the behavior that we
want, and I'll try to give an explanation of why. But please
bear with me... I'm still a beginner at understanding some
of the theory behind numerical computing, so I may make some
mistakes. Hopefully those who understand it better will
correct me.
It's a fact that only certain numbers can be represented exactly
when using numbers of finite precision. For instance, floating
point numbers where the exponent is larger than the number of
digits of precision can represent certain integers but not
others. The same situation arises in the use of floating (or
fixed) point decimals -- only certain fractions can be expressed
exactly.
There are (at least) two different mental models we can use to
deal with this limitation of finite precision. One (which you
are espousing above) is to treat each limited-precision
number as if it had an inherent uncertainty, and it represented
all those numbers which are closer to that representable
number than any other. The other mental model is to suppose
that each limited-precision number represents just one
specific value and that the other values simply cannot be
represented.
I always understand things better when I have concrete examples,
so let me use numbers in base 10 with a precision of 3 to
illustrate these two viewpoints. With a precision of 3 we can
write out the values 2.34x10^4 for the number 23,400 and
2.35x10^4 for the number 23,500. But 23,401, 23,402, and all the
others in between can't be represented. So by the "Error bar"
mental model, we think of "2.34x10^4" as shorthand for "some
number from 23,350 to 23,450". By the "Exact values" mental
model we think of "2.34x10^4" as meaning "The exact value
23,340".
Now that we know there are two different mental models, we
have to wonder whether one is inherently better than the
other (and whether it makes any difference). I maintain that
it DOES make a difference, and that the "Exact values" mental
model is BETTER than the "Error bar" model. The reason why it
is better doesn't come into play until you start performing
operations on the numbers, so let's examine that.
Suppose that you are adding two numbers under the "Exact
values" mental model. Adding 1.23x10^4 and 3.55x10^4 is quite
easy... the exact sum of 12,300 and 35,500 is 47,800 or
4.78x10^4. But if we add 1.23x10^4 and 8.55x10^2, then the
"real" sum is 12,300 + 855 = 13,155. That CAN'T be represented
in the "exact values" mental model, so we have to ROUND IT
OFF and store "1.32x10^4" instead. In general, the values are
quite clear in the "Exact values" model, but OPERATIONS may
require rounding to express the result.
Proponents of the "Error bar" mental model are now thinking
"yes, well our model trades complexity of VALUES for
simplicity in OPERATIONS". But unfortunately that's just not
true. Again, let us try to add 1.23x10^4 and 3.55x10^4. What
that REALLY means (in the "Error bar" mental model) is that
we are adding SOME number in the range 12,250..12,350 to SOME
number in the range 35,450..35,550. The result must be some
number in the range 47,700..47,900. That is LIKELY to belong
to the range 47,750..47,850 (ie 4.78x10^4), but it MIGHT also
come out to something in the 4.77x10^4 range or the 4.79x10^4
range. We intent to return a result of 1.78x10^4, so operations
in the "Error bar" mental model aren't simple, but contain
some sort of "rounding off" as well, only the exact rules for
how THIS "rounding off" works are a bit more confusing -- I'm
not sure I could express them very well.
(Side Note: You might think to fix this problem by saying that
instead of perfect ranges, the numbers represent probability
curves centered around the given number. By using gaussian
curves, you might even be able to make addition "simple" (at
the cost of a very complex idea of "values"). But whatever
distribution worked for addition would NOT work for
multiplication or other operations, so this trick is doomed
to failure.)
So it turns out that WHATEVER you do, operations on finite-
precision numbers are going to require some sort of a "rounding
off" step to keep the results within the domain. That being
said, the "Exact values" mental model is superior. Not only
does it allow a very simple interpretation of the "values",
but it *also* allows the definition of "rounding off" to be
part of the definition of the operations and not be linked to
the values themselves. (For instance, I prefer a simple
"round-to-nearest" rule, but someone else may have a different
rule and that only affects how they perform operations.) It
is for this reason that (I think) most serious specifications
for the behavior of finite-precision numbers prefer to use
the "Exact value" mental model.
(Another side note: You will notice that the argument here
is strikingly similar to the behavior of PEP 327's Decimal
type. That's because I learned just about everything I'm saying
here from following the discussions (and doing the recommended
reading) about Facundo's work there.)
Okay, in my long-winded fashion, I may have shown that the
"Exact values" mental model is superior to the "Error bars"
model. But does it really make any difference to anyone other
than writers of numerical arithmetic specifications? It does,
and one of the best examples is the one that started this
conversation... comparing instances of different numerical
types. I maintain that when comparing a long with a float
where the exponent is larger than the precision, that the
float should be treated as if it were EXACTLY EQUAL TO
<coefficient>*2^<exponent>, instead of trying to treat it as
some sort of a range. Similarly, if we implemented a Rational
type, I would suggest that instances of float (or of Facundo's
Decimal) where <exponent> is LESS than the digits of
<coefficient> should be treated as if they were EXACTLY EQUAL
TO the corresponding fraction-over-a-power-of-two.
All right, those who ACTUALLY understand this stuff (rather
than having learned it from this newsgroup) can go ahead and
correct me now.
Once-a-teacher,-always-trying-to-give-a-lecture-lly yours,
-- Michael Chermside
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